Question

Erika was working on solving the exponential equation 50x = 17; however, she is not quite sure where to start. Using complete sentences, describe to Erika how to solve this equation and how solving would be different if the bases were equal. (10 points)

Answer 1

50^x

Answer 2

Erika could solve with a calculator and do trial and error until she found some power of 5 that would equal 17 (it's a little above 1.76 by this method). But being a smart person, Erika would see that the best solution would be to take logs of both sides

Log 5^x = log 17....... Erika knows that log 5^x = x(log 5)
x(log 5) = log 17
0.69897x = 1.2304489....divide across by 0.69897
x = 1.76055, a much more precise solution

With the above you can surely write a few sentences.

Answer 3

i saw that on yahoo answers but i dont think my teacher would like the trial and error part

Answer 4

It's ok ill use that for that question but how about this one Brett has determined a function f(x) that shows the exponential growth of the number of shoes Larae owns each year. Explain how the f-1(x) can be found and what f-1(132) means. (10 points)

Answer 5

@dan815

Answer 6

and for the first one there's the equal bases part

Answer 7

@jim_thompson5910

Answer 8

`Erika was working on solving the exponential equation` \(\LARGE 50^x = 17\) `; however, she is not quite sure where to start. Using complete sentences, describe to Erika how to solve this equation and how solving would be different if the bases were equal.`

use logs to isolate exponents

example

\[\large 2^x = 10 \implies x = \log_{\ 2}(10) = \frac{\log(10)}{\log(2)} \approx 3.3219\]

Answer 9

I used the change of base formula to get the approximate decimal form

Answer 10

so would i write x=log50(17)

Answer 11

yeah \[\LARGE x = \log_{50}(17)\]

then you use the change of base formula to get the approximate value of x

Answer 12

0.724

Answer 13

Rules:

\[\Large b^x = y \rightarrow x = \log_b(y)\]

Change of base formula
\[\Large \log_{b}\left(x\right)=\frac{\log\left(x\right)}{\log\left(b\right)}\]

Answer 14

I'm getting 0.724 as well

Answer 15

so that means \[\LARGE 50^{0.724} \approx 17\]

Answer 16

yay thanks! what about the second part? like if the bases were equal

Answer 17

if the bases are equal, then you can set the exponents equal and solve for x

Answer 18

thanks so much you're the best! Dan who??

Answer 19

example

\[\Large 2^3 = 2^{x+1}\]

the bases are both 2, so the exponents must be equal

therefore, 3 = x+1

Answer 20

ohh okay i get it

Answer 21

what about for the brett problem? Brett has determined a function f(x) that shows the exponential growth of the number of shoes Larae owns each year. Explain how the f-1(x) can be found and what f-1(132) means. (10 points)

Answer 22

by " f-1(x)" you mean \(\LARGE f^{-1}(x)\) right?

Answer 23

yes

Answer 24

what does that notation mean? any ideas?

Answer 25

no i dont understand it

Answer 26

it means "inverse function of f"

Answer 27

the inverse undoes whatever operation was applied

so say you add initially, the inverse would be subtraction
if you multiply, the inverse is division
if you square something, the inverse is the square root

Answer 28

so it can be found depending on what has been done?

Answer 29

so would f-1(132) be 132

Answer 30

-132

Answer 31

what undoes exponents?

Answer 32

LOGS

Answer 33

YOU TAUGHT ME THAT

Answer 34

yes you will use logs to get the inverse of f

we can't actually find the inverse since we don't know what the function f is

Answer 35

so for what it means would i just write it means the inverse of 132? or the inverse of f of 132?

Answer 36

the original f(x) function takes an x value, which is the number of years, and produces a y value

y = f(x)

in goes x ----> out comes y or f(x)

x = number of years
y = number of shoes

Answer 37

the inverse takes everything in reverse because we're undoing everything

with the inverse, the y value is now the input, the x is the output

in goes y into the inverse ----> out comes x

Answer 38

why is this important? because we can use the inverse to answer questions like "in what year will the number of shoes be 132?"

Answer 39

so it would be right if i wrote that f-1(132) means that you take the inverse of it and now the y value is the input and the x is the output?

Answer 40

yeah

Answer 41

yaya thankyouuuu